# Tagged Questions

A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

20 views

### Probabilistic exponential growth model

I have a real valued number $y_t$. At each time step t, $y_t$ is multiplied by $(1 + \epsilon)$ with probability $p$ and multiplied by $(1 - \epsilon)$ with probability $1 - p$. What is the expected ...
12 views

### Derivation of first passage time density for Ornstein-Uhlenbeck Process

Consider the following OU process parameterized by two positive real numbers $\alpha$ and $\sigma$: $$dX_t -\alpha X_t dt + \sigma \sqrt{2\alpha} dW_t$$ $$X_0 = x_o > 0$$ We are interested in the ...
14 views

13 views

### Ergodicity in hidden Markov models

Assume that we have a hidden Markov model, where we have a sequence of hidden variables $Z_1, \dots, Z_m$ which form a Markov chain. Now, at each "time point" $i$, an observation $Y_i$ is drawn from ...
45 views

### Suggestions for Constructing a Random Variables from Correlated Observations

Let $\mathcal{X} \neq \phi$ be a finite set. Let $P_{XY_1Y_2}$ be a fixed joint distribution over $\mathcal{X}\times\mathcal{X}\times\mathcal{X}\$ and that a random sample $(X,Y_1,Y_2 )$ is drawn ...
62 views

### Ornstein-Uhlenbeck Process simulation bug

I think I found a bug in a programm somebody posted but I can't fix it. It is about the simulation of an Ornstein-Uhlenbeck Process. The problem is from this [article][1] & and from wikipedia from ...
17 views

47 views

27 views

### Probability density function of Poisson Process trajectory

Given a Poisson Process with rate $\lambda$, by a fixed time $t$ we have observed $n$ arrivals at times $t_1 < \cdots < t_n$, with $t_0 = 0 < t_1$ and $t_n < t$ I'm trying to find a ...
47 views

### What Stochastic Calculi Other Than Ito And Stratonovich Exist?

When learning about stochastic calculus, you typically encounter Ito and Stratonovich calculi, usually in that order. There are many differences between the two (Ito processes have better martingale ...
23 views

### Stochastic Differential Equation Time-Independent

I know that a generic 1-D SDE can be written in Ito form as: $dX_{t} = \mu(X_t,t)dt + \sigma(X_t,t)dW_t$. I was curious as to how such an SDE is written when modelling time-independent processes. I ...
41 views

### Expected hitting time of Ornstein-Uhlenbeck process

If I recall correctly, it is known that for a standard brownian motion starting at $0$, that the expected time to hit some level $a>0$ is infinite. I'm curious if there's a proof of what the ...
73 views

### Is the integral of an Ito process still an Ito process?

Assume $r(t)$ is an Ito diffusion: $$dr_t = \mu_tdt + \sigma_tdW_t$$ Then, define the following process: $$X_t = \int_0^tr(s)ds$$ Is $X_t$ still an Ito diffusion?
34 views

### proof of Markov chain Monte Carlo

This is the first step of proof of MCMC in my notes I have a question, how come $\pi(x)\pi(x_p\mid x)=\pi(x_p)\pi(x\mid x_p)$? Is it true for any markov chains which are ergodic and aperiodic? The ...
22 views